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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# MTH9879 Homework 2\n",
      "\n",
      "Assigned: February 10, 2015.\n",
      "Due: February 24, 2015 by 6pm. \n",
      "\n",
      "Late homework **will not be accepted**.\n",
      "\n",
      "$$\n",
      "\\newcommand{\\supp}{\\mathrm{supp}}\n",
      "\\newcommand{\\E}{\\mathbb{E} }\n",
      "\\def\\Cov{{ \\mbox{Cov} }}\n",
      "\\def\\Var{{ \\mbox{Var} }}\n",
      "\\newcommand{\\1}{\\mathbf{1} }\n",
      "\\newcommand{\\PP}{\\mathbb{P} }\n",
      "%\\newcommand{\\Pr}{\\mathrm{Pr} }\n",
      "\\newcommand{\\QQ}{\\mathbb{Q} }\n",
      "\\newcommand{\\RR}{\\mathbb{R} }\n",
      "\\newcommand{\\DD}{\\mathbb{D} }\n",
      "\\newcommand{\\HH}{\\mathbb{H} }\n",
      "\\newcommand{\\spn}{\\mathrm{span} }\n",
      "\\newcommand{\\cov}{\\mathrm{cov} }\n",
      "\\newcommand{\\sgn}{\\mathrm{sgn} }\n",
      "\\newcommand{\\HS}{\\mathcal{L}_{\\mathrm{HS}} }\n",
      "%\\newcommand{\\HS}{\\mathrm{HS} }\n",
      "\\newcommand{\\trace}{\\mathrm{trace} }\n",
      "\\newcommand{\\LL}{\\mathcal{L} }\n",
      "%\\newcommand{\\LL}{\\mathrm{L} }\n",
      "\\newcommand{\\s}{\\mathcal{S} }\n",
      "\\newcommand{\\ee}{\\mathcal{E} }\n",
      "\\newcommand{\\ff}{\\mathcal{F} }\n",
      "\\newcommand{\\hh}{\\mathcal{H} }\n",
      "\\newcommand{\\bb}{\\mathcal{B} }\n",
      "\\newcommand{\\dd}{\\mathcal{D} }\n",
      "\\newcommand{\\g}{\\mathcal{G} }\n",
      "\\newcommand{\\p}{\\partial}\n",
      "\\newcommand{\\half}{\\frac{1}{2} }\n",
      "\\newcommand{\\T}{\\mathcal{T} }\n",
      "\\newcommand{\\bi}{\\begin{itemize}}\n",
      "\\newcommand{\\ei}{\\end{itemize}}\n",
      "\\newcommand{\\beq}{\\begin{equation}}\n",
      "\\newcommand{\\eeq}{\\end{equation}}\n",
      "\\newcommand{\\beas}{\\begin{eqnarray*}}\n",
      "\\newcommand{\\eeas}{\\end{eqnarray*}}\n",
      "\\newcommand{\\cO}{\\mathcal{O}}\n",
      "\\newcommand{\\cF}{\\mathcal{F}}\n",
      "\\newcommand{\\cL}{\\mathcal{L}}\n",
      "\\newcommand{\\BS}{\\text{BS}}\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<font color = \"red\">Homework is to be done by each student individually.  To receive full credit, you must email a completed copy of this iPython notebook to Yu Gan (yugan323@gmail.com), Fubo Shi (fubo.shi.baruch@gmail.com), and Tai-Ho Wang (tai-ho.wang@baruch.cuny.edu) by the due date and time.  All R-code must run correctly and solutions must be written up neatly in Markdown/LaTeX format.\n",
      "\n",
      "<font color=\"blue\">If you encounter problems with Markdown/LaTeX or iPython notebook, please contact your TAs Yu Gan and/or Fubo Shi.\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### 0. (0 points) \n",
      "\n",
      "Read Chapter 12 of Hasbrouck and Section 6.4 of Foucault, Pagano and R&ouml;ell.\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## The Parlour (1998) model"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### 1. (10 points) \n",
      "Consider the first step $t = 1$ of the 3-step order book case presented in the lecture slides. Suppose that the order book at (or just before) $t = 1$ has one share offered and one share bid ($n_B = n_A = 1$).\n",
      "\n",
      "Give the probabilities as of a market buy, market sell, limit buy and limit sell at $t = 1$,\n",
      "You may use my Mathematica notebook *Parlour.nb* as a guide (if you wish)."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### Solution:\n",
      "\n",
      "The probability of a limit buy order at $t=1$ being filled is $\\PP = \\frac{5}{22} \\times \\frac{5}{22}=\\frac{25}{484}.$\n",
      "\n",
      "Traders will submit a MS only if\n",
      "$B - \\beta_t\\,V \\geq (A - \\beta_t\\,V) \\,\\PP$ or\n",
      "equivalently\n",
      "\n",
      "$$\\beta_1 \\leq \\frac{B-A\\,\\PP}{V\\,(1-\\PP)} = \\frac{B-A\\,\\PP}{V\\,(1-\\PP)} =\\frac{4540}{5049}.$$\n",
      "\n",
      "Thus,\n",
      "\\begin{eqnarray*}\n",
      "{\\mathbb{P} }_{MB}&=&{\\mathbb{P} }_{MS}=\\frac{1}{2}{\\mathbb{P} }\\left(\\beta_1<\\frac{4540}{5049}\\right)=\\frac{1135}{5049}\\\\\n",
      "{\\mathbb{P} }_{LB}&=&{\\mathbb{P} }_{LS}=\\frac{1}{2}{\\mathbb{P} }\\left(\\frac{4540}{5049}<\\beta_1<\\frac{12}{11}\\right)=\\frac{22}{459}\\\\\n",
      "{\\mathbb{P} }_{0}&=&1-2{\\mathbb{P} }_{MB}-2{\\mathbb{P} }_{LB}=1-\\frac{2270}{5049}-\\frac{44}{459}=\\frac{5}{11}.\\end{eqnarray*}"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### 2. (4 points) \n",
      "Explain why the optimal market/limit order decision depends in general on quantity available on both sides of the order book in the Parlour model."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### Solution:\n",
      "The probability of the Limit Order Probability depends on the order book for the next time step, which is related to the quantitity available on both sides. And the market order probability depends on the probability of Limit order\n",
      "    "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Set up R-environment"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%load_ext rmagic"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "download.file(url=\"http://mfe.baruch.cuny.edu/wp-content/uploads/2015/01/ziSetup.zip\", destfile=\"ziSetup.zip\")\n",
      "unzip(zipfile=\"ziSetup.zip\")\n",
      "source(\"ziSetup.R\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "trying URL 'http://mfe.baruch.cuny.edu/wp-content/uploads/2015/01/ziSetup.zip'\n",
        "Content type 'application/zip' length 1485 bytes\n",
        "opened URL\n",
        "==================================================\n",
        "downloaded 1485 bytes\n",
        "\n"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Cont-Kukanov optimal order placement"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### 3. (16 points)\n",
      "\n",
      "Consider the two BATS exchanges BZX and BYX.  Suppose you are trying to buy shares of MSFT in 3.9 minutes ($1\\%$ of the trading day).  Suppose further that the two exchanges have the following characteristics:\n",
      "\n",
      "**Exchange** | **Make fee** | **Take fee** | **Daily volume** \n",
      "-----|-----:|-------:| ---- |\n",
      "| **(per share)** | **(per share)** | **(shares)** \n",
      "BZX | -0.32\u00a2 | +0.30\u00a2 | 10,500,000\n",
      "BYX | +0.03\u00a2 | -0.03\u00a2 | 1,300,000\n",
      "\n",
      "Assume the daily volatility of MSFT is $1\\%$ and that MSFT is trading at $\\$37.50$.\n",
      "\n",
      "(a) We may suppose that if your order is only partially filled, that's because the price moved away (*adverse selection*).  This suggests that we penalize any under fill of limit orders by $\\sigma\\,\\sqrt{T}$ where $T$ is the order duration.  Compute the corresponding value of $\\lambda_u$ in cents per share.\n",
      "\n",
      "\n",
      "(b) Assuming a spread of $1$ cent, what are $h$, $f$, and $r$ for BZX and BYX respectively (in cents per share)?\n",
      "\n",
      "\n",
      "(c) Assuming no seasonality, what is the mean value $\\bar \\xi$ of $\\xi$ (over 3.9 minutes) for BZX and BYX respectively (in shares)?\n",
      "\n",
      "\n",
      "(d)  Assume that $\\xi$ is log normally distributed with\n",
      "\n",
      "$$\n",
      "\\xi = \\bar \\xi \\,\\exp\\left\\{\\sigma\\,Z-\\frac 12\\,\\sigma^2\\right\\}.\n",
      "$$\n",
      "\n",
      "where $Z \\sim N(0,1)$ and $\\sigma = 1/2$.  What is the standard deviation of $\\xi$ (in terms of $\\bar \\xi$)?\n",
      "\n",
      "\n",
      "(e) Write R-code to determine the optimal order size $L^\\star+Q$ on BZX and BYX respectively. Assume here that you will either trade on BZX or BYX; there is no splitting of the order between the two exchanges. <br>\n",
      "(Hint: You may find the R function *qnorm(. )* useful).\n",
      "\n",
      "\n",
      "(f) Compare the optimal limit order sizes from part (e) with the ratios of quote sizes and volumes on BZX and BYX.  Are your results from part (e) roughly consistent with observation.  You may assume that the quote sizes shown on the MSFT BATS slides from Lecture 1 are typical."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### Solution:\n",
      "\n",
      "(a)$\\ \\lambda_u = \\sigma\\,\\sqrt{T} = 1\\% \\times 37.50 \\times \\sqrt{1\\%} \\times 100 = 3.75\\ $  \n",
      "\n",
      "(b)\n",
      "For BZX: $h=0.5$, $f=0.3$, $r=-0.32$. \n",
      "For BYX: $h=0.5$, $f=-0.03$, $r=0.03$.\n",
      "\n",
      "(c)\n",
      "For BZX: $\\bar \\xi = 1\\% \\times 10500000 = 105000$. \n",
      "For BYX: $\\bar \\xi = 1\\% \\times 1300000 = 13000$.\n",
      "\n",
      "(d) Since $Z \\sim N(0,1)$ and $\\sigma = 1/2$, then $\\sigma\\,Z-\\frac 12\\,\\sigma^2 \\sim N(-0.125,0.25)$. Thus the standard deviation of $\\xi$ is $\\bar\\xi\\sqrt{e^\\frac 14 - 1} \\approx 0.5329\\ \\bar\\xi$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "lamda <- 3.75\n",
      "h <- 0.5\n",
      "r <- 0.32\n",
      "f <- 0.3\n",
      "xi <- 1050000\n",
      "sigma <- 0.5\n",
      "tmp <- (2 * h + f + r) / (lamda + h + r)\n",
      "op_order <- qnorm(tmp)\n",
      "op_order <- xi * exp(op_order * sigma - 0.5 * sigma * sigma)\n",
      "print(op_order)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "[1] 768873.6\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "lamda <- 3.75\n",
      "h <- 0.5\n",
      "r <- -0.03\n",
      "f <- -0.03\n",
      "xi <- 130000\n",
      "sigma <- 0.5\n",
      "tmp <- (2 * h + f + r) / (lamda + h + r)\n",
      "op_order <- qnorm(tmp)\n",
      "op_order <- xi * exp(op_order * sigma - 0.5 * sigma * sigma)\n",
      "print(op_order)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "[1] 78340.38\n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "(f) Rather roughly consistent."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## The Bouchaud-M&eacute;zard-Potters order book approximation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### 4. (16 points)\n",
      "\n",
      "When $\\lambda(u)=\\alpha$,  a constant, the BMP order density becomes\n",
      "\n",
      "\\begin{eqnarray*}\n",
      "\\rho_{BMP}(\\Delta) &=& \\frac{\\alpha}{\\delta}\\,\\left\\{e^{-\\Delta/{\\tilde \\Delta}}\\,\\int_0^{\\Delta/{\\tilde\\Delta}}\\,\\sinh(u)\\,du + \\sinh(\\Delta/{\\tilde \\Delta})\\,\\int_{\\Delta/{\\tilde\\Delta}}^\\infty\\,e^{-u}\\,du\\right\\} \\\\\n",
      "&=&\\frac{\\alpha}{\\delta}\\,\\left\\{1 - e^{-\\Delta/{\\tilde \\Delta}}  \\right\\}\n",
      "\\end{eqnarray*}\n",
      "\n",
      "where the characteristic distance\n",
      "\n",
      "$$\n",
      "\\tilde \\Delta=\\sqrt{\\frac{\\sigma^2}{2\\,\\delta}}.\n",
      "$$\n",
      "\n",
      "(a) Amend the SFGK zero intelligence code from Problem 3 of HW1 to count the number of limit orders generated in addition to computing the average book shape.\n",
      "\n",
      "(b) With $\\alpha = 1$, $\\mu = 10$ and $\\delta = \\frac15$, simulate 100,000 events and plot the resulting average book shape in red (in a band $\\pm 20$ around the mid price).\n",
      "\n",
      "(c) From your simulation, what is the probability $P_L$ of a limit order at a given level conditional on an event? (Recall that there are 60 limit order levels in the simulation).\n",
      "\n",
      "(d) In [Gatheral and Oomen]<sup id=\"cite_ref-GO\" class=\"reference\"><a href=\"#cite_note-GO\"><span>[</span>1<span>]</span></a></sup>, with these SFGK parameters, realized variance per event was estimated as\n",
      "\n",
      "$$\n",
      "\\sigma^2_{GO} = 0.083.\n",
      "$$\n",
      "\n",
      "Using your estimate $P_L$ of limit orders per level per event, convert this estimate to variance per limit order (recall that $\\alpha=1$).  Denote your result by $\\sigma^2_{GO}$.  Compute the characteristic distance $\\tilde \\Delta_{GO}$ with this estimate $\\sigma^2_{GO}$.\n",
      "\n",
      "(e) In Lecture 1, using dimensional analysis, we estimated that variance $\\sigma^2_{DA}$ per unit time should be of the order of\n",
      "\n",
      "$$\n",
      "\\sigma^2_{DA} = \\frac{\\mu\\,\\delta}{\\alpha^2}.\n",
      "$$\n",
      "\n",
      "Compute the characteristic distance $\\tilde \\Delta_{DA}$ with this estimate $\\sigma^2_{DA}$.\n",
      "\n",
      "(f)  Using your two estimates of $\\tilde \\Delta$ from parts (d) and (e),  write functions to compute the corresponding order book densities $\\rho_{GO}(\\Delta)$ and $\\rho_{DA}(\\Delta)$.\n",
      "\n",
      "(g)\n",
      "Superimpose plots of the asymptotic densities $\\rho_{DA}(\\Delta)$ (in blue) and $\\rho_{DA}(\\Delta)$ (in orange) on your plot of the average book shape."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "\n",
      "#Generate an event and update the buy and sell books\n",
      "#Note that limit orders may be placed inside the spread\n",
      "generateEvent_cos <- function()\n",
      "    {\n",
      "    nb <<- sum(book$buySize[book$Price>=(bestOffer()-L)]); # Number of cancelable buy orders\n",
      "    ns <<- sum(book$sellSize[book$Price<=(bestBid()+L)]); # Number of cancelable sell orders\n",
      "    eventRate <- nb*delta + ns*delta + mu + 2*L*alpha;\n",
      "    probEvent <- c(L*alpha,L*alpha,nb*delta,ns*delta,mu/2,mu/2)/eventRate;\n",
      "    m <- sample(1:6, 1, replace = TRUE, probEvent); #Choose event type\n",
      "    switch(m,\n",
      "    \t\tlimitBuyOrder(),\n",
      "    \t\tlimitSellOrder(),\n",
      "    \t\tcancelBuyOrder(),\n",
      "    \t\tcancelSellOrder(),\n",
      "    \t\tmarketBuyOrder(),\n",
      "    \t\tmarketSellOrder()\n",
      "    \t\t);\n",
      "    return (m)\n",
      "    }"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "logging <- F # Very important for speed!\n",
      "\n",
      "alpha <- 1\n",
      "mu <- 10\n",
      "delta <- 1/5 \n",
      "initializeBook5()\n",
      "\n",
      "count_limit_order=0\n",
      "count_total=0\n",
      "numEvents <- 100000 # Average over 100,000 events\n",
      "avgBookShape_I <- bookShape(20)/numEvents\n",
      "for(count in 2:numEvents){\n",
      "  m=generateEvent_cos()\n",
      "    if (m==1) {count_limit_order=count_limit_order+1}\n",
      "    if (m==2) {count_limit_order=count_limit_order+1}\n",
      "    count_total=count_total+1\n",
      "  avgBookShape_I <- avgBookShape_I+bookShape(20)/numEvents\n",
      "\n",
      "}\n",
      "\n",
      "plot(-20:20,avgBookShape_I,main=\"avgBookShape_I\",xlab=\"Relative price\",ylab=\"# Shares\", col=\"red\", type=\"b\")\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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3bqhSBXPnIjsb4eHo1090ICOyezdeeIG3fzQpLGjStiZN0KSJ6BDGaPlyHl1n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\n"
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%R\n",
      "cat(\"count_total: \", count_total)\n",
      "cat(\"count_limit_order: \", count_limit_order)\n",
      "\n",
      "PL=count_limit_order/60/count_total\n",
      "cat(\"PL: \", PL)\n",
      "\n",
      "i=-30\n",
      "Variance=0\n",
      "for (count in 1:61)\n",
      "{\n",
      "    Variance= Variance+PL*i*i\n",
      "    i=i+1\n",
      "}\n",
      "Variance=Variance/60/30\n",
      "cat(\"Variance: \", Variance)\n",
      "\n",
      "delta_go=sqrt(Variance/0.4)\n",
      "cat(\"delta_go\", delta_go)\n",
      "\n",
      "Variance_DA=10/5/30\n",
      "delta_DA=sqrt(Variance_DA/0.4)\n",
      "cat(\"delta_DA: \", delta_DA)\n",
      "\n",
      "rou_go <- function(delta)\n",
      "{\n",
      "    return (5*(exp(1-delta/delta_go)))\n",
      "}\n",
      "rou_DA <- function(delta)\n",
      "{\n",
      "    return (5*(exp(1-delta/delta_DA)))\n",
      "}\n",
      "\n",
      "delta=0\n",
      "y_go<-array(1:100)\n",
      "y_DA<-array(1:100)\n",
      "i=1\n",
      "for(count in 1:100){\n",
      "y_go[i]=rou_go(delta)\n",
      "y_DA[i]=rou_DA(delta)\n",
      "delta=delta+0.02\n",
      "i=i+1\n",
      "}\n",
      "x=0.02*(1:100)\n",
      "\n",
      "plot(x,y_go,main=\"Rou\",xlab=\"delta\",ylab=\"Rou\", col=\"blue\", type=\"b\")\n",
      "lines(x,y_DA,col=\"orange\")\n",
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "text": [
        "count_total:  99999count_limit_order:  49998PL:  0.008333083Variance:  0.08754367delta_go 0.4678239delta_DA:  0.4082483"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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f4+ScBwdu3sTe/v6ivT0pKUqHEkKoSwpaP5Rux+VwMu6/T6VhQ375Je/nu3dZ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\nJiISKAY0EZFA/R+VnVtCtox8bQAAAABJRU5ErkJggg==\n"
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### 5. (6 points)\n",
      "\n",
      "Another of Mike and Farmer's observations is that the distribution of marketable as well as unmarketable limit orders as a function of distance $\\Delta$ to the same-side best quote is well-described by a Student-$t$ distribution.  Assume $\\mu=1/2$ as above.\n",
      "\n",
      "(a) Derive an expression in terms of the Student-$t$ cumulative density function for the ratio of arrival rates of limit to market orders as a function of the spread $s$.  How does this ratio vary with $s$?\n",
      "\n",
      "(b) Relative to the zero-intelligence model, does this empirically observed strategic behavior make the spread revert faster or more slowly?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Solution:\n",
      "(a)\n",
      "$$f(x)=\\frac{\\Gamma(\\frac{\\nu+1}{2})}{\\sqrt{\\nu\\,\\pi}\\,\\Gamma(\\frac{\\nu}{2})}\\,\\frac{1}{\\left(1+\\frac{x^2}{\\nu}\\right)^{\\frac{\\nu+1}{2}}}\\sim \\frac{1}{s^{\\nu+1}}\\text{ as }$$\n",
      "which is a power function of s\n",
      "\n",
      "\n",
      "(b) In the ZI model, orders rates are constant. But here the orders rates are smaller as the spread goes up. So the revert will be faster."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### References\n",
      "\n",
      "<div class=\"reflist\" style=\"list-style-type: decimal;\">\n",
      "\n",
      "<ol>\n",
      "\n",
      "  <li id=\"cite_note-GO\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-GO\">^</a></b></span> Jim Gatheral and Roel C. A Oomen, Zero-intelligence realized variance estimation, <span>*Finance and Stochastics*,</span> <span>**14**</span>(2) 249\u2013283, 2010.</li>\n",
      "\n",
      "</ol>"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}